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Rosenbrock—Wanner–Type Methods pp 49–68Cite as

Exponential Rosenbrock Methods and Their Application in Visual Computing

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Part of the Mathematics Online First Collections book series (MOFC)

Abstract

We introduce a class of explicit exponential Rosenbrock methods for the time integration of large systems of stiff differential equations. Their application with respect to simulation tasks in the field of visual computing is discussed where these time integrators have shown to be very competitive compared to standard techniques. In particular, we address the simulation of elastic and nonelastic deformations as well as collision scenarios focusing on relevant aspects like stability and energy conservation, large stiffnesses, high fidelity and visual accuracy.

Keywords

  • Accurate and efficient simulation
  • (Explicit) exponential Rosenbrock integrators
  • Stiff order conditions
  • Stiff elastodynamic problems
  • Visual computing

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Notes

  1. 1.

    We estimated the error after 60 s of simulated time based on the accumulated Euclidean distances of the individual particles in the position space compared to ground truth values which are computed with a sufficiently small step size.

  2. 2.

    In order to detect collisions efficiently, we make use of a standard bounding volume hierarchy.

References

  1. A.H. Al-Mohy, N.J. Higham, Computing the action of the matrix exponential, with an application to exponential integrators. SIAM J. Sci. Comput. 33, 488–511 (2011)

    CrossRef  MathSciNet  Google Scholar 

  2. U. Ascher, S. Ruuth, B. Wetton, Implicit-explicit methods for time-dependent PDEs. SIAM J. Numer. Anal. 32(3), 797–823 (1997)

    CrossRef  Google Scholar 

  3. D. Baraff, A. Witkin, Large steps in cloth simulation, in ACM Transactions on Graphics, SIGGRAPH’98 (ACM, New York, 1998), pp. 43–54

    Google Scholar 

  4. M. Bergou, M. Wardetzky, S. Robinson, B. Audoly, E. Grinspun, Discrete elastic rods. ACM Trans. Graph. 27(3), 63:1–63:12 (2008)

    Google Scholar 

  5. M. Caliari, A. Ostermann, Implementation of exponential Rosenbrock-type integrators. Appl. Numer. Math. 59(3–4), 568–581 (2009)

    CrossRef  MathSciNet  Google Scholar 

  6. M. Caliari, P. Kandolf, A. Ostermann, S. Rainer, The Leja method revisited: backward error analysis for the matrix exponential. SIAM J. Sci. Comput. 38(3), A1639–A1661 (2016)

    CrossRef  MathSciNet  Google Scholar 

  7. W.L. Chao, J. Solomon, D. Michels, F. Sha, Exponential integration for Hamiltonian Monte Carlo, in Proceedings of the 32nd International Conference on Machine Learning, ed. by F. Bach, D. Blei. Proceedings of Machine Learning Research, vol. 37, pp. 1142–1151 (PMLR, Lille, 2015)

    Google Scholar 

  8. Y.J. Chen, U.M. Ascher, D.K. Pai, Exponential Rosenbrock-Euler integrators for elastodynamic simulation. IEEE Trans. Visual. Comput. Graph. 24(10), 2702–2713 (2018)

    CrossRef  Google Scholar 

  9. S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems. J. Comput. Phys. 176(2), 430–455 (2002)

    CrossRef  MathSciNet  Google Scholar 

  10. C. Curtiss, J.O. Hirschfelder, Integration of stiff equations. Proc. Natl. Acad. Sci. 38(3), 235–243 (1952)

    CrossRef  MathSciNet  Google Scholar 

  11. C.F. Curtiss, J.O. Hirschfelder, Integration of stiff equations. Proc. Natl. Acad. Sci. USA 38(3), 235–243 (1952)

    CrossRef  MathSciNet  Google Scholar 

  12. B. Eberhardt, O. Etzmuß, M. Hauth, Implicit-explicit schemes for fast animation with particle systems, in Proceedings of the 11th Eurographics Workshop on Computer Animation and Simulation (EGCAS) (Springer, Berlin, 2000), pp. 137–151

    Google Scholar 

  13. S. Gaudreault, J. Pudykiewicz, An efficient exponential time integration method for the numerical solution of the shallow water equations on the sphere. J. Comput. Phys. 322, 827–848 (2016)

    CrossRef  MathSciNet  Google Scholar 

  14. C. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice–Hall, Englewood Cliffs, 1971)

    Google Scholar 

  15. S. Geiger, G. Lord, A. Tambue, Exponential time integrators for stochastic partial differential equations in 3D reservoir simulation. Comput. Geosci. 16(2), 323–334 (2012)

    CrossRef  Google Scholar 

  16. R. Goldenthal, D. Harmon, R. Fattal, M. Bercovier, E. Grinspun, Efficient simulation of inextensible cloth, in ACM Transactions on Graphics, SIGGRAPH’07 (2007)

    Google Scholar 

  17. M.A. Gondal, Exponential Rosenbrock integrators for option pricing. J. Comput. Appl. Math. 234(4), 1153–1160 (2010)

    CrossRef  MathSciNet  Google Scholar 

  18. E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems (Springer, New York, 1996)

    CrossRef  Google Scholar 

  19. M. Hauth, O. Etzmuss, A high performance solver for the animation of deformable objects using advanced numerical methods. Comput. Graph. Forum 20, 319–328 (2001)

    CrossRef  Google Scholar 

  20. N.J. Higham, Functions of Matrices: Theory and Computation (SIAM, Philadelphia, 2008)

    CrossRef  Google Scholar 

  21. M. Hochbruck, C. Lubich, On Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal. 34, 1911–1925 (1997)

    CrossRef  MathSciNet  Google Scholar 

  22. M. Hochbruck, A. Ostermann, Explicit exponential Runge–Kutta methods for semilinear parabolic problems. SIAM J. Numer. Anal. 43, 1069–1090 (2005)

    CrossRef  MathSciNet  Google Scholar 

  23. M. Hochbruck, A. Ostermann, Explicit integrators of Rosenbrock-type. Oberwolfach Rep. 3, 1107–1110 (2006)

    Google Scholar 

  24. M. Hochbruck, A. Ostermann, Exponential integrators. Acta Numer. 19, 209–286 (2010)

    CrossRef  MathSciNet  Google Scholar 

  25. M. Hochbruck, C. Lubich, H. Selhofer, Exponential integrators for large systems of differential equations. SIAM J. Sci. Comput. 19, 1552–1574 (1998)

    CrossRef  MathSciNet  Google Scholar 

  26. M. Hochbruck, A. Ostermann, J. Schweitzer, Exponential Rosenbrock-type methods. SIAM J. Numer. Anal. 47, 786–803 (2009)

    CrossRef  MathSciNet  Google Scholar 

  27. S. Krogstad, Generalized integrating factor methods for stiff PDEs. J. Comput. Phys. 203(1), 72–88 (2005)

    CrossRef  MathSciNet  Google Scholar 

  28. M.W. Kutta, Beitrag zur näherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435–453 (1901)

    MATH  Google Scholar 

  29. V.T. Luan, Fourth-order two-stage explicit exponential integrators for time-dependent PDEs. Appl. Numer. Math. 112, 91–103 (2017)

    CrossRef  MathSciNet  Google Scholar 

  30. V.T. Luan, A. Ostermann, Exponential B-series: the stiff case. SIAM J. Numer. Anal. 51, 3431–3445 (2013)

    CrossRef  MathSciNet  Google Scholar 

  31. V.T. Luan, A. Ostermann, Explicit exponential Runge–Kutta methods of high order for parabolic problems. J. Comput. Appl. Math. 256, 168–179 (2014)

    CrossRef  MathSciNet  Google Scholar 

  32. V.T. Luan, A. Ostermann, Exponential Rosenbrock methods of order five–construction, analysis and numerical comparisons. J. Comput. Appl. Math. 255, 417–431 (2014)

    CrossRef  MathSciNet  Google Scholar 

  33. V.T. Luan, A. Ostermann, Stiff order conditions for exponential Runge–Kutta methods of order five, in Modeling, Simulation and Optimization of Complex Processes - HPSC 2012, H.B. et al. (ed.) (Springer, Berlin, 2014), pp. 133–143

    Google Scholar 

  34. V.T. Luan, A. Ostermann, Parallel exponential Rosenbrock methods. Comput. Math. Appl. 71, 1137–1150 (2016)

    CrossRef  MathSciNet  Google Scholar 

  35. V.T. Luan, J.A. Pudykiewicz, D.R. Reynolds, Further development of the efficient and accurate time integration schemes for meteorological models J. Comput. Sci. 376, 817–837 (2018)

    MathSciNet  MATH  Google Scholar 

  36. D.L. Michels, M. Desbrun, A semi-analytical approach to molecular dynamics. J. Comput. Phys. 303, 336–354 (2015)

    CrossRef  MathSciNet  Google Scholar 

  37. D.L. Michels, J.P.T. Mueller, Discrete computational mechanics for stiff phenomena, in SIGGRAPH ASIA 2016 Courses (2016), pp. 13:1–13:9

    Google Scholar 

  38. D.L. Michels, G.A. Sobottka, A.G. Weber, Exponential integrators for stiff elastodynamic problems. ACM Trans. Graph. 33(1), 7:1–7:20 (2014)

    Google Scholar 

  39. D.L. Michels, J.P.T. Mueller, G.A. Sobottka, A physically based approach to the accurate simulation of stiff fibers and stiff fiber meshes. Comput. Graph. 53B, 136–146 (2015)

    CrossRef  Google Scholar 

  40. D.L. Michels, V.T. Luan, M. Tokman, A stiffly accurate integrator for elastodynamic problems. ACM Trans. Graph. 36(4), 116 (2017)

    Google Scholar 

  41. J. Niesen, W.M. Wright, Algorithm 919: a Krylov subspace algorithm for evaluating the φ-functions appearing in exponential integrators. ACM Trans. Math. Softw. 38, 3 (2012)

    CrossRef  MathSciNet  Google Scholar 

  42. D.A. Pope, An exponential method of numerical integration of ordinary differential equations. Commun. ACM 6, 491–493 (1963)

    CrossRef  MathSciNet  Google Scholar 

  43. C.D. Runge, Über die numerische Auflösung von Differentialgleichungen. Math. Ann. 46, 167–178 (1895)

    CrossRef  MathSciNet  Google Scholar 

  44. A. Stern, M. Desbrun, Discrete geometric mechanics for variational time integrators, in SIGGRAPH 2006 Courses (2006), pp. 75–80

    Google Scholar 

  45. A. Stern, E. Grinspun, Implicit-explicit variational integration of highly oscillatory problems. Multiscale Model. Simul. 7, 1779–1794 (2009)

    CrossRef  MathSciNet  Google Scholar 

  46. A. Tambue, I. Berre, J.M. Nordbotten, Efficient simulation of geothermal processes in heterogeneous porous media based on the exponential Rosenbrock–Euler and Rosenbrock-type methods. Adv. Water Resour. 53, 250–262 (2013)

    CrossRef  Google Scholar 

  47. D. Terzopoulos, J. Platt, A. Barr, K. Fleischer, Elastically deformable models. ACM Trans. Graph. 21, 205–214 (1987)

    Google Scholar 

  48. M. Tokman, J. Loffeld, P. Tranquilli, New adaptive exponential propagation iterative methods of Runge-Kutta type. SIAM J. Sci. Comput. 34, A2650–A2669 (2012)

    CrossRef  MathSciNet  Google Scholar 

  49. H. Zhuang, I. Kang, X. Wang, J.H. Lin, C.K. Cheng, Dynamic analysis of power delivery network with nonlinear components using matrix exponential method, in 2015 IEEE Symposium on Electromagnetic Compatibility and Signal Integrity (IEEE, Piscataway, 2015), pp. 248–252

    Google Scholar 

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Acknowledgements

The first author has been partially supported by NSF grant DMS–2012022. The second author has been partially supported by King Abdullah University of Science and Technology (KAUST baseline funding).

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Mississippi State University, Starkville, MS, USA

    Vu Thai Luan

  2. Computational Sciences Group, Visual Computing Center, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia

    Dominik L. Michels

Authors
  1. Vu Thai Luan
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Corresponding author

Correspondence to Vu Thai Luan .

Editor information

Editors and Affiliations

  1. Fachbereich Elektrotechnik. Maschinenbau und Technikjournalismus, Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Nordrhein-Westfalen, Germany

    Dr. Tim Jax

  2. Arbeitsgruppe Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Dr. Andreas Bartel

  3. Arbeitsgruppe Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Prof. Dr. Matthias Ehrhardt

  4. Arbeitsgruppe Angewandte Mathematik und Numerische Analysis, Bergische Universität Wuppertal, Wuppertal, Nordrhein-Westfalen, Germany

    Prof. Dr. Michael Günther

  5. Fachbereich Elektrotechnik. Maschinenbau und Technikjournalismus und Institut für Technik, Ressourcenschonung und Energieeffizienz, Hochschule Bonn-Rhein-Sieg, Sankt Augustin, Nordrhein-Westfalen, Germany

    Prof. Dr. Gerd Steinebach

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Luan, V.T., Michels, D.L. (2021). Exponential Rosenbrock Methods and Their Application in Visual Computing. In: Jax, T., Bartel, A., Ehrhardt, M., Günther, M., Steinebach, G. (eds) Rosenbrock—Wanner–Type Methods. Mathematics Online First Collections. Springer, Cham. https://doi.org/10.1007/978-3-030-76810-2_3

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